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Elliptic Loading
Created by: Brandon Sforzo __TOC__ Elliptic Loading is a force distribution on a wing that has an elliptical form in the spanwise direction. It is well known that this load distribution results in the least induced drag for a given wing span, and total lift. Though elliptic planforms naturally result in elliptic load distributions, there are other methods to design for this reduced drag advantage. Elliptic Lift Distribution The force distribution across a wing taking an elliptical form is one where the lift per unit span, L' measured is a function of the spanwise distance from the centerline, y . The function is defined by, : L'=L_0'\sqrt{1-\left ( \cfrac{y}{b/2} \right )^2} Where L_0' is the magnitude of the lift at the centerline, and b is the span length.Shevell,R. (1989). Fundamentals of Flight. Englewood Cliffs: Prentice-Hall. Through substitution of the circulation for the elliptical distribution into the equation for the induced drag coefficient, the induced drag is found to be, : C_{D,i}=\cfrac{C_L^2}{\pi AR} This is the same as the general induced drag equation, : C_{D,i}=\cfrac{C_L^2}{\pi e AR} if the efficiency factor, e=1 .Anderson,J.(2005).''Fundamentals of Aerodynamics.''McGraw-Hill This illustrates that the induced drag for a wing with an elliptic loading for the same aspect ratio and span is at a minimum. Characteristics An elliptic lift distribution causes the downwash across the span to be a constant. As a result the induced drag is a minimum when compared to a planar wing of equal span, total lift and velocity. The elliptic loading can be achieved by use of an elliptic planform with no wing twist. The same distribution can also be achieved by adding the appropriate twist to the wing, altering the local angle of attack for a given chord of the wing. For the elliptic planform wing with no twist, since the absolute angle of attack and downwash angle are spanwise constant, the effective angle of attack is also constant. The defining characteristics of an ellpitical loading are: *For a planar wing, the induced drag is at a minimum, for a fixed span, total lift, and equivalent velocity. *The downwash across the span is constant. *The elliptic lift distribution occurs naturally on a planar wing with an elliptical planform. The angle of attack for all airfoil sections of the wing are the same for the untwisted (planar) wing. *The effective angle of attack, \alpha_E , and the local lift coefficient, c_l , are constant along the span as a result of the absolute angle of attack, \alpha_a , and the downwash angle, \epsilon , are constant along the span. Elliptic Distribution Restoration Observing that an elliptic planform produces an elliptical lift distribution it could be expected that a rectangular or tapered planform produces a trapezoidal lift distribution. As it turns out, deviations away from an elliptical distribution cause downwash effects across the wing that tend to restore the elliptic distribution. For a rectangular wing with taper the loading profile is between the profile of the wing and the corresponding elliptical distribution for similar span. The simple approximation of calculating lift distributions by taking the midway function between the wing profile and the elliptical distribution is a good first order approximation.Schrenk,O.''A Simple Approximation Method for Obtaining Spanwise Lift Distribution.''NACA TM 1910, 1940. Washout The wing twist to decrease lift at the tips and increase at the roots is called washout. The angle of attack at a particular spanwise position can be chosen to cause the lift profile to be elliptical. Washout is also advantageous because the higher geometric angles of attack at the roots of the wings allow that area to stall before the wing tips. Aillerons are placed closer to the wing tips than the roots to have a higher moment, and control is maintained through a partial stall because of the washout. Planforms The Supermarine Spitfire (pictured at the top of this article), is the classic example of an aircraft with an elliptical planform, Though seen clearly in the picture the planform is not an exact ellipse. The planform is a combination of two halves of different characteristic ellipses. The spitfire design requirements called for a very thin wing thickness, yet a thicker root so the landing gear could retract. The wings also needed to house the weaponry of the aircraft as well as the fuel. According to the designers, the ellpitically shaped wings allowed for these requirements to be met. The benefits from the elliptic loading were secondary in the design.Glancey,J.(2006).''Spitfire: The Biography.''Atlantic, p. 36. Another example of an eliptical planform, even more true to the shape than the spitfire, is the Heinkel He 70. It was suspected that the spitfire design originated from the Heinkel. Structure Elliptic planforms, though providing elliptic load distributions, are structurally complex. The internal ribbing of the airfoil structure would need to conform to the rounded shape, requiring more bracing. This adversely removes space for fuel in the wing. Large wings, or those with large aspect ratios are also disadvantaged by the complex structure required for an elliptic planform. These aircraft larger than small fighters alternately choose a rectangular or swept wing with a geometric twist to adjust the load distribution. References